Step 1: Identify the wave number from the given wave equation. The general equation for a progressive wave is $y = A\sin(\omega t - kx)$, where $k$ is the wave number. The given equation is $y = 0.1\sin (400\pi t - 20\pi x)$. By comparing the two equations, we can see that the wave number $k = 20\pi \text{ rad/m}$. Step 2: Use the formula for phase difference. The phase difference $\Delta\phi$ between two points separated by a distance $\Delta x$ is given by the formula: $$\Delta\phi = k \Delta x$$ We are given $\Delta x = 0.05 \text{ m}$. Step 3: Substitute the values and calculate the phase difference. $$\Delta\phi = 20\pi \text{ rad/m} \times 0.05 \text{ m}$$ $$\Delta\phi = 20\pi \times \frac{5}{100}$$ $$\Delta\phi = 20\pi \times \frac{1}{20}$$ $$\Delta\phi = \pi \text{ radians}$$ Step 4: Compare the result with the given options. The calculated phase difference is $\pi$ radians, which corresponds to option B. The correct option is B. $\boxed{\text{B. } \pi \text{ radians}}$ That's 2 down. 3 left today — send the next one.